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Wang Tiles for Image and Texture Generation. Michael Cohen Microsoft Research Jonathan Shade University of Washington and WildTangent Stefan Hiller University of Constance Oliver Deussen University of Constance. Wang Tiles. Rules Tilings are assembled by matching like edges

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Wang Tiles for Image and Texture Generation

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Wang Tiles for Image and Texture Generation

Michael CohenMicrosoft Research

Jonathan ShadeUniversity of Washington

and WildTangent

Stefan HillerUniversity of Constance

Oliver DeussenUniversity of Constance


Wang Tiles

  • Rules

    • Tilings are assembled by matching like edges

    • Tiles cannot be rotated

  • In 1961 Wang asserted that no aperiodic set of tiles exists, i.e., a set that

    • COULD tile the plane

    • could NOT tile it periodically

  • He was wrong, but still got to keep his name on them

  • Aperiodic sets can be used to

    • Simulate Turing machines

    • Act as life forms in science fiction (“Wang’s Carpets” by Greg Egan)


Wang Tiles – not new to CG

  • Jos Stam

    • Showed the use of aperiodic tile sets for texture assembly


Wang Tiles – not new to CG

  • Jos Stam

    • Showed the use of aperiodic tile sets for texture assembly

  • Neyret and Cani

    • Used triangular tiles with coded edges to assure continuous texture


Wang Tiles

Wang Tiles

Mirrored


Wang Tiles


Wang Tiles

  • We introduce non-periodic tilings

    • set can trivially tile the plane periodically

    • simple stochastic algorithm → non-periodic

non-periodic

aperiodic


Wang Tiles

  • 2 colors = 8 tiles

  • 3 colors = 18 tiles

  • C colors ?

2 x C2


Why Wang Tiles?

  • Once you fill a set of Wang Tiles with stuff

  • You can create as much of that stuff as you want for almost free.

Free stuff!

Free stuff!

Demo


Wang Tiles for Texture Synthesis


Wang Tiles for Texture Synthesis


Wang Tiles for Texture Synthesis


Wang Tiles for Texture Synthesis


Wang Tiles for Texture Synthesis


Wang Tiles for Texture Synthesis

Demo


Wang Tiles for Texture Synthesis


?

Wang Tiles: the corner problem


Wang Tiles: the corner problem


Wang Tiles: the corner problem

A better solution? Lefebvre and Neyret, I3D 2003


Wang Tiles for Distributions

Poisson Disc Distributions


Wang Tiles for Distributions


Wang Tiles for Distributions


Wang Tiles for Distributions


Wang Tiles for Distributions


Wang Tiles for Distributions


Wang Tiles for Distributions


Wang Tiles for Distributions


Wang Tiles for Distributions


Wang Tiles for Distributions


3D Wang Tiles

  • Volumetric tiles filled with 3D “texture”

  • In this example, 2.5D

  • Layered Depth Image samples


3D Wang Tiles

  • Hierarchical

  • View Dependent


3D Wang Tiles


Wang Tiles

  • Simple stochastic assembly: non-periodic

  • Texture synthesis

  • Distribution synthesis

  • The corner problem

  • Some 3D applications

Free stuff!

Free stuff!


Wang Tiles

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Don Mitchell

Alex Colburn

Reviewers


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